# Groups of order 56 with Sylow 2-subgroup isomorphic $Q_8$

I try to classify non-abelian groups of order $$56$$ with sylow $$2$$-subgroup isomorphic to quaterion group $$Q_8$$. More accurately I want to construct $$2$$ non-isomorphic such groups. This is an excercise 5.3.7 from Dummit and Foote's book. I can construct such groups as semidirect product as follows $$G_1 = C_7 \rtimes_{\phi_{1}} Q_8$$, $$G_2 = C_7 \rtimes_{\phi_{2}} Q_8$$, $$G_3 = C_7 \rtimes_{\phi_{3}} Q_8$$, where $$\phi_n: Q_8 \to \operatorname{Aut}(C_7)$$. Let $$Q_8 = $$ and $$<\sigma>$$ be a unique subgroup of order $$2$$ of $$\operatorname{Aut}(C_7) \cong C_6$$, where $$\sigma$$ inverts elements of $$C_7$$. We define $$\phi_1$$ as $$\phi_1(i) = \sigma, \phi_1(j) = \operatorname{id}$$ (here $$\operatorname{id}$$ is an identical automorphism), $$\phi_2(i) = \operatorname{id}, \phi_2(j) = \sigma$$ and $$\phi_3(i) = \sigma, \phi_3(j) = \sigma$$

My question. Am I right that all this groups are isomorphic? And if all this groups are isomorphic what is the second type of such gruops?

I use the following result about isomorphism of semidirect product. Let $$\tau\in \operatorname{Aut}(H)$$ than $$N\rtimes_{\phi} H \cong N\rtimes_{\tau\phi} H$$.

You are correct: those are all isomorphic. The second type is of course the direct product. That is different as $$C_7$$ is central in the direct product of $$C_7$$ with $$Q_8$$.